Question

[Computer] In general, the analysis of coupled oscillators with dissipative forces is much more complicated than the conservative case considered in this chapter. However, there are a few cases where the same methods still work, as the following problem illustrates: (a) Write down the equations of motion corresponding to (11.2) for the equal-mass carts of Section 11.2 with three identical springs, but with each cart subject to a linear resistive force \(-b \mathbf{v}\) (same coefficient \(b\) for both carts). (b) Show that if you change variables to the normal coordinates \(\xi_{1}=\frac{1}{2}\left(x_{1}+x_{2}\right)\) and \(\xi_{2}=\frac{1}{2}\left(x_{1}-x_{2}\right),\) the equations of motion for \(\xi_{1}\) and \(\xi_{2}\) are uncoupled. (c) Write down the general solutions for the normal coordinates and hence for \(x_{1}\) and \(x_{2}\). (Assume that \(b\) is small, so that the oscillations are underdamped.) (d) Find \(x_{1}(t)\) and \(x_{2}(t)\) for the initial conditions \(x_{1}(0)=A\) and \(x_{2}(0)=v_{1}(0)=v_{2}(0)=0,\) and plot them for \(0 \leq t \leq 10 \pi\) using the values \(A=k=m=1,\) and \(b=0.1\).

Short answer

The equations of motion transform into uncoupled forms using normal coordinates. Initial conditions give the specific solution values.

Step-by-step solution

Understanding the Damped System

We're building on a simple coupled oscillator system with each cart also subject to a linear resistive force (-bv). This adds a damping term to our problem.

Write down the Equations of Motion

For the system with equal masses and damping, the equations of motion for each cart are:\(m\ddot{x}_1 = -k x_1 + k (x_2 - x_1) - b\dot{x}_1\)\(m\ddot{x}_2 = -k x_2 + k (x_1 - x_2) - b\dot{x}_2\)

Change Variables to Normal Coordinates

Introduce normal coordinates: \( \xi_1 = \frac{1}{2}(x_1 + x_2) \) and \( \xi_2 = \frac{1}{2}(x_1 - x_2) \). Replace \(x_1\) and \(x_2\) in the equations of motion with these new variables.

Derive Uncoupled Equations

The change of variables transforms the system into two separate equations:\((m\ddot{\xi}_1) = -b\dot{\xi}_1\)\((m\ddot{\xi}_2) = -2k\xi_2 - b\dot{\xi}_2\)

Solve for the General Solutions

Solving each uncoupled equation yields:\(\xi_1(t) = C_1 e^{-\frac{b}{m}t}\)\(\xi_2(t) = e^{-\frac{b}{2m}t}(C_2 \cos \omega_d t + C_3 \sin \omega_d t)\), where \(\omega_d = \sqrt{2k/m - (b/2m)^2}\) assuming underdamping.

Find Expressions for x_1 and x_2

Using the expressions for \(\xi_1\) and \(\xi_2\), solve for \(x_1\) and \(x_2\):\(x_1(t) = \xi_1(t) + \xi_2(t)\)\(x_2(t) = \xi_1(t) - \xi_2(t)\)

Apply Initial Conditions

Given the initial conditions \(x_1(0) = A\) and \(x_2(0) = v_1(0) = v_2(0) = 0\), resolve constants in the general solution:\(C_1 = \frac{A}{2},\ C_2 = \frac{A}{2},\ C_3 = 0\).

Plot x_1(t) and x_2(t)

Using parameters \(A = k = m = 1\) and \(b = 0.1\), plot \(x_1(t)\) and \(x_2(t)\) for the range \(0 \leq t \leq 10\pi\). Calculations give explicit time-dependent functions for \(x_1(t)\) and \(x_2(t)\) based on original formulas and resolved constants.

Key concepts

Damped Oscillations

In the study of coupled oscillators, damping is an essential factor that describes the reduction in amplitude of oscillations over time. A damped oscillation is one that experiences a force opposite to its velocity, often referred to as a resistive or frictional force. This is mathematically represented by a damping term \(-b\dot{v}\), where \(b\) is the damping coefficient, and \(\dot{v}\) is the velocity of the oscillator.

Damping occurs due to various reasons such as air resistance or internal friction within the material of the oscillator. There are different types of damping:

• Underdamping: The system oscillates with a progressively decreasing amplitude.
• Critical damping: The system returns to equilibrium without oscillating.
• Overdamping: The system returns to equilibrium without oscillating but more slowly than in the case of critical damping.

For small values of the damping coefficient \(b\), the system is underdamped, meaning that it will continue to oscillate with gradually decreasing amplitude. This can be seen in coupled oscillators subject to light damping.

Equations of Motion

The equations of motion are fundamental mathematical expressions that describe the dynamics of a physical system under the influence of forces. For coupled oscillators, these equations incorporate terms for the spring force and damping force acting on each mass.

In the example of equal-mass carts connected by springs, each cart is subject to a resistive force \(-b\dot{x}\). The equations of motion for the carts are derived from Newton's second law, which states that the net force on an object is equal to the mass times its acceleration \(\,m\ddot{x}\). For our coupled oscillator system, the equations are:

• For cart1: \(m\ddot{x}_1 = -k x_1 + k (x_2 - x_1) - b\dot{x}_1\)
• For cart2: \(m\ddot{x}_2 = -k x_2 + k (x_1 - x_2) - b\dot{x}_2\)

These equations reflect that both carts are connected by springs creating restoring forces dependent on their displacements. The damping terms \(-b\dot{x}_1\) and \(-b\dot{x}_2\) illustrate the resistant force affecting each cart's motion.

Normal Coordinates

Normal coordinates are a powerful tool in simplifying the analysis of coupled oscillations. By transforming the system's original coordinates to normal coordinates, we can decouple the equations of motion, making them easier to solve.

For a system of two coupled oscillators with coordinates \(x_1\) and \(x_2\), the normal coordinates are defined as:

• \(\xi_1 = \frac{1}{2}(x_1 + x_2)\)
• \(\xi_2 = \frac{1}{2}(x_1 - x_2)\)

These transformations create two separate equations: one for \(\xi_1\) and another for \(\xi_2\), effectively uncoupling the system's motion. The new equations are possibly simpler, each describing a separate, independent motion. In this particular case, the transformations yield:

• \(m\ddot{\xi}_1 = -b\dot{\xi}_1\)
• \(m\ddot{\xi}_2 = -2k\xi_2 - b\dot{\xi}_2\)

This separation allows for easier calculation of each of the normal modes of oscillation of the system.

Underdamped Systems

An underdamped system is characterized by oscillations that gradually decay over time due to the presence of damping. In technical terms, the damping is weak enough that the system continues to oscillate, but the amplitude of these oscillations decreases exponentially.

In mathematical analysis, for an underdamped oscillator, the solution typically includes sinusoidal terms multiplied by an exponential decay factor. For the normal coordinates, this results in specific expressions:

• \(\xi_1(t) = C_1 e^{-\frac{b}{m}t}\)
• \(\xi_2(t) = e^{-\frac{b}{2m}t}(C_2 \cos \omega_d t + C_3 \sin \omega_d t)\)

Here, \(\omega_d\) is the damped natural frequency given by \(\omega_d = \sqrt{2k/m - (b/2m)^2}\), showing that the frequency is slightly lower than the natural frequency due to the damping. Since \(b = 0.1\) is small, the system oscillates under this slightly modified frequency, demonstrating the underdamped nature.

The phenomenon is often observed in mechanical systems like vehicle suspension and electrical circuits, where oscillations are desirable but should be lightly controlled to prevent excessive motion or noise.